Number of muffins that can be eaten in three hours. 4 variables.

Question

Assuming that all clones have the same appetite, in one hour, $100$ clones can eat $1500$ cupcakes and $1000$ muffins. In two hours, $ 60 $ clones can eat $1200$ cupcakes and $1500$ muffins. In three hours, $50$ clones can eat $750$ cupcakes and $x $muffins. Find $x$.

I tried using variations here by letting the number of clones be equal to the number of cupcakes to be eaten times a variable $k$ all over the number of hours times the number of muffins to be eaten, but $k$ does not seem to be constant.

Answer 1

Hmm, I wanted to only give you hints, but now I went and wrote down the full answer... But at least I didn't put that much explanation, so you still need to do some work to understand the solution below. ;)

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Let $c$ be standing for cupcakes, $m$ for muffins and $a$ for the appetite of a single clone per hour. Then you have the following equations:

$\begin{align*} 100a &= 1500c + 1000m \\ 2\cdot 60a &= 1200c + 1500m \\ 3\cdot 50a &= 750c + xm \end{align*}$

Putting this into a matrix, we get $$A = \begin{pmatrix} -100 & 1500 & 1000 \\ -120 & 1200 & 1500 \\ -150 & 750 & x\end{pmatrix}.$$

This matrix should have a non-trivial null space. (why?) How to check if the null space is non-trivial? Simple: The determinant has to be zero. The determinant of that thing is $60000x - 135000000$, so $x = 2250$.

Answer 2

The first relation says that in one hour one clone has the appetite for $15$ cupcakes and $10$ muffins. Let the appetite required to eat a cupcake be $a$ and that for a muffin be $b$, then assuming a clone has unit appetite, $15a+10b=1$.

The second relation then says that in one hour $60$ clones can eat $600$ cupcakes and $750$ muffins, or one clone can eat $10$ cupcakes and $\frac{25}2$ muffins, or $10a+\frac{25}2b=1$. Solving the linear system gives $b=\frac2{35}$ and $a=\frac1{35}$.

The final relation says that in one hour $50$ clones can eat $250$ cupcakes and $\frac x3$ muffins, or one clone can eat $5$ cupcakes and $\frac x{150}$ muffins, or $5a+\frac x{150}b=1$. Substituting the $a$ and $b$ we found gives $\frac17+\frac{2x}{150×35}=1$, or $x=75×5×6=2250$.